balitsky–kovchegov evolution equation · introduction • when the center of mass energy in a...

Balitsky–Kovchegov evolution equation

Emmanuel Gr ave de [email protected]

High Energy Phenomenology Group

Instituto de Fısica

Universidade Federal do Rio Grande do Sul

Porto Alegre, Brazil

GFPAE – UFRGShttp://www.if.ufrgs.br/gfpae

BK, E.G. de Oliveira–GFPAE – p. 1

Outline

• Introduction• Dipole picture• Balitsky–Kovchegov equation• Solutions to BK equation• Saturation scale• GBW model• BK equation in momentum space• Traveling waves• NLO BK• Pomeron loops• Fluctuations


Introduction

• When the center of mass energy in a collision is much bigger than the fixed hardscale (for example, in DIS, the photon virtuality is the hard scale and s≫ Q2),parton densities increase with growing energy.

• This produces larger scattering amplitudes.

• In this regime of saturation, BFKL equation is not valid anymore.

• Using the qq-dipole model, Kovchegov was able to derive a new equation.

• Kovchegov equation is not exact, but it is a mean field approximation of a hierarchyof equations derived previously by Balitsky.

• This equation is nonlinear and can be reduced to BFKL equation when thenonlinear term is disregarded.


Dipole picture

• For simplicity, we should consider deep inelastic scattering of a virtual photon on ahadron or nucleus.

• In the dipole picture, the system is studied in the reference frame of the target (thehadron or the nucleus is at rest).

• Therefore, all QCD evolution is included in the wave function of the incomingphoton.

• The incoming photon cannot directly strongly interact with the target.

• The splitting of the photon into a quark-antiquark pair is then considered.

• This heavy quark-antiquark pair is called onium.

• Quarks have color, but the incident photon is colorless.

• Then, the quark has color and the antiquark the correspondent anticolor.


Dipole picture

• In the figure, the photon splitting is shown.

• z1 is the fraction of light cone momentum.

• x0 and x1 are quark and antiquark positions (x01 = x0 − x1).


Gluon emission

• The quark or antiquark can emit soft gluons (z2 << z1).

• These soft gluons can also emit softer gluons.


Planar diagrams

• Each loop count as Nc.

• Each vertex count as αs.

• Therefore, planar diagrams are enhanced by N2c compared to nonplanar diagrams

of the same order em αs.


Large-Nc limit

• In the limit of large number of colors (Nc → ∞), these soft gluons can beconsidered as quark-antiquark pairs.

• Therefore, the original dipole (x01) splits into two new dipoles (x02 and x21).


The onium

• The original incoming photon can be represented now by an arbitrary number ofdipoles.

• These dipoles do not interact with each other.


Onium-hadron scattering

• Now, the nucleus or hadron is not going to scatter off the photon, but off thecomplex structure of color dipoles:

pp

• Two gluons must be exchanged to conserve color.


Single and multiple scattering

• Single dipole scattering leads to linear BFKL equation.

pp

• Multiple dipole scatterings lead to nonlinear BK equation.

pp


Fan diagrams

• Pomeron merging in the target can be understood as a multiple scattering betweentarget and projectile.

≡

• Therefore, all fan diagrams are included.γ

N

*


Diagrams not included

• Not all diagrams are included, however.

• Diagrams representing Pomeron merging (a) cannot be represented as a multiplescattering (b).

AAA

(a)

* *

(b)

γγ


F2 structure function

• In the dipole picture, the F2 structure function is given by:

F2(x,Q2) =Q2

4π2αem

Z

d2x01dz

2πΦ(x01, z)σdip(x01, Y ) (1)

where Φ(x01, z) = ΦT (x01, z) + ΦL(x01, z) is the sum of the square of thetransverse (T) and longitudinal (L) photon wavefunctions (more specifically, thewavefunction for a photon to go into a dipole)

ΦT (x01, z) =2NcαEM

πa2K2

1 (x01a)[z2 + (1 − z)2] (2)

ΦL(x01, z) = 4Q2z2(1 − z)2K20 (x01a). (3)

with a2 = Q2z(1 − z).

• σdip is the dipole-nucleus (or nucleon) cross section


The forward scattering amplitude

• The dipole cross section is given by

σdip(x01, Y ) = 2

Z

d2b01N (b01,x01, Y ). (4)

• The quantity b01 is the impact parameter given by:

b01 =x1 + x0

2(5)

• N (b01,x01, Y ) is the propagator of the quark-antiquark pair through the nucleusrelated to the forward scattering amplitude of the quark-antiquark pair with thenucleus.


Balitsky–Kovchegov equation

• The Balitsky–Kovchegov equation is given by:

dN (b01,x01, Y )

dY=

αs

2π

Z

d2x2x2

01

x220x

212

h

N“

b01 +x21

2,x20, Y

”

+N“

b01 +x20

2,x21, Y

”

−N (b01,x01, Y )

−N“

b01 +x12

2,x20, Y

”

N“

b01 +x20

2,x21, Y

”i

(6)

• This equation evolves N (b01,x01, Y ).

• The evolution quantity is the rapidity Y ≈ ln1/x.

• Besides the evolution variable, one has 4 degrees of freedom (2 from b01 and 2from x01).

• It resumms all powers of αsln1/x.

• αs = αsNc/π


Balitsky approach

• A general feature of high-energy scattering is that a fast particle moves along itsstraight-line classical trajectory and the only quantum effect is the eikonal phasefactor acquired along this propagation path.

• In QCD, for the fast quark or gluon scattering off some target, this eikonal phasefactor is a Wilson line:

• the infinite gauge link ordered along the straight line collinear to particle’s velocitynµ:

Uη(x⊥) = Pexpn

ig

Z ∞

−∞du nµ A

µ(un+ x⊥)o

, (7)

• Here Aµ is the gluon field of the target, x⊥ is the transverse position of the particlewhich remains unchanged throughout the collision, and the index η labels therapidity of the particle.

• Repeating the argument for the target (moving fast in the spectator’s frame) wesee that particles with very different rapidities perceive each other as Wilson lines.

• Therefore, Wilson-line operators form the convenient effective degrees of freedomin high-energy QCD.


Balitsky approach

• At small xB = Q2/(2p · q), the virtual photon decomposes into a pair of fastquarks moving along straight lines separated by some transverse distance.

• The propagation of this quark-antiquark pair reduces to the “propagator of thecolor dipole” U(x⊥)U†(y⊥) - two Wilson lines ordered along the direction collinearto quarks’ velocity.

• The structure function of a hadron is proportional to a matrix element of this colordipole operator

Uη(x⊥, y⊥) = 1 − 1

NcTr{Uη(x⊥)U†η(y⊥)} (8)

• sandwiched by target’s states.

• The gluon parton density is approximately

xBG(xB , µ2 = Q2) ≃ 〈p| Uη(x⊥, 0)|p〉

˛

˛

˛

x2⊥

=Q−2(9)

where η = ln 1xB

.


Balitsky approach

• The energy dependence of the structure function is translated then into thedependence of the color dipole on the slope of the Wilson lines determined by therapidity η.

• At relatively high energies and for sufficiently small dipoles we can use the leadinglogarithmic approximation (LLA) where αs ≪ 1, αs lnxB ∼ 1 and get thenon-linear BK evolution equation for the color dipoles:

d

dηU(x, y) =

αsNc

2π2

Z

d2z(x− y)2

(x− z)2(z − y)2[U(x, z)+ U(y, z)−U(x, y)−U(x, z)U(z, y)]

(10)


Balitsky hierarchy

• The operator equation creates an hierarchy of equations for the observables:

d

dη

D

U(x, y)E

=αsNc

2π2

Z

d2z(x− y)2

(x− z)2(z − y)2

hD

U(x, z)E

+D

U(y, z)E

−D

U(x, y)E

−D

U(x, z)U(z, y)Ei

(11)

d

dη〈Txy〉 =

αs

2π

Z

d2z M(x, y, z) [〈Txz〉 + 〈Tyz〉 − 〈Txy〉 − 〈TxzTzy〉] (12)

• In the mean field approximation, 〈TxzTzy〉 = 〈Txz〉〈Tzy〉, and BK equation isrecovered.


The BFKL limit

• The last (and nonlinear) term is a direct consequence of taking into accountmultiple scatterings.

• BK equation is a closed equation, instead of Balitsky original hierarchy ofequations.

• The mean field approximation of Balitsky first equation gives the BK equation.

• If we disregard the last term, we obtain the following linear equation that can beshown equivalent to BFKL equation:

dN (b01,x01, Y )

dY=

αs

2π

Z

d2x2x2

01

x220x

212

h

N“

b01 +x21

2,x20, Y

”

+N“

b01 +x21

2,x20, Y

”

−N (b01,x01, Y )i

(13)


BK versus AGL equation

• BK, AGL and GLR equations sum the same diagrams (for example):

• All three use the eikonal approximation, i. e., incoming photon momentum haslarge q+.

• However, BK equation does not consider large Q2. It resums all terms1/(2q+q−) ≈ 1/Q2.


BK in 0+1 dimensions

• If one neglects the spatial dependence (N (x02,b0, Y )), BK equation is given bythe logistic equation:

dNdY

= ω(N −N 2) ω > 0. (14)

• This equation has two fixed points, one unstable (N = 0) and another stable(N = 1).

• For every initial condition in 0 < N ≤ 1, for Y → ∞, N → 1.

• In other words, there is saturation for highY (small x).

• If the N 2 term is neglected, the fixed point at N = 1 disappears and N → ∞ forY → ∞.



• The logistic equation complete solution is (C = N (Y = 0)):

N (Y ) =CeωY

1 + C(eωY − 1). (15)

• Solutions: linear (red) and nonlinear (blue) equations.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10

C=0.01

Y

N(Y

)



• If impact parameter is disregarded and only sizes of dipoles are considered:

N (b,x, Y ) → N (r, Y ). (16)

• BK equation is then simplified:

dN (|x01|, Y )

dY=

αs

2π

Z

d2x2x2

01

x220x

212

[N (|x20|, Y ) + N (|x21|, Y )

−N (|x01|, Y ) −N (|x21|, Y )N (|x20|, Y )] (17)

• Physically speaking, this represents the scattering on infinite and uniform nucleus.

• We note that the kernel already depends only on dipole sizes.



• The same kind of solutions of BK equation in (0+1) dimensions is found for BKequation in (1+1) dimensions for fixed r.

• Solutions: linear (red) and nonlinear (blue) equations.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2 4 6 8 10 12 14 16 18

r=0.48

r=4.8

Y

N(Y

,r)



• Solutions for different initial conditions in linear (above) and log (below) scales.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5 6 7 8 9 10r=|x01|

N(Y

,r)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5 6 7 8 9 10r=|x01|

N(Y

,r)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5 6 7 8 9 10r=|x01|

N(Y

,r)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10-2

10-1

1 10r=|x01|

N(Y

,r)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10-2

10-1

1 10r=|x01|

N(Y

,r)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

10-2

10-1

1 10r=|x01|

N(Y

,r)


Saturation scale

• The figures shown previously can be divided into three regions in r:

◦ The small r region, where nonlinear corrections are negligible;

◦ The large r region, where nonlinear corrections dominate and N ≈ 1;

◦ The region between the first two.

• One can introduce the saturation scale:

r <1

Qs(Y )→ N << 1,

r >1

Qs(Y )→ N ≈ 1.


Saturation scale

• One can get a saturation scale from BK solutions.

• Neglecting some corrections, it is given by:

Qs(Y ) = Q0exp(αsλY )Y −β λ ≈ 2.4

• The saturation scale is the dividing line between dense and dilute regions.

• The higher the rapidity the denser the system gets and partons start to re-interact.

• Also, saturation occurs earlier if the size of partons is bigger.


Saturation scale

• Dilute and saturated regimes can be represented in 2-D graphs (Y = ln(1/x)):

Q1r= −

Y

sQ (Y)

Q1r= −

Y

Non−perturbative

Saturation scale

Dense system

region

Dilute systemLinear evolution

Nonlinear evolution


GBW model

• Linear growing and saturation properties of BK equation are roughly similarlyfound in the Golec-Biernat and Wusthoff (GBW) saturation model.

• One of the key ideas to GBW model is the assumption of a x-dependent radius:

R0(x) =1

Q0

„

x

x0

«λ

2

, (18)

which scales the quark-antiquark separation r in the dipole cross section:

r =r

2R0(x). (19)

• Then, the dipole cross section can be written as:

σdip(x, r2) = σ0g(r2). (20)

• Q0 = 1 GeV sets the dimension.


GBW model: dipole cross section

• The function g(r2) chosen is:

g(r2) = 1 − exp[−r2]. (21)

• The dipole cross section is then:

σdip(x, r) = σ0

1 − exp

"

−„

r

2R0

«2#!

= σ0

„

1 − exp

»

−r2Q20

4

“x0

x

”λ–«

(22)

• The three parameters were fitted to σ0 = 23mb, λ = 0.288 and x0 = 310−4.

• The cross section can be written as a function of Y also:

σdip(Y, r) =

Z

d2bN(b, r, Y ) = σ0

»

1 − exp

„

− r2Q2

s(Y )

4

«–

with Q2s(Y ) = exp(0.28(Y − Y0)).


GBW model: σdip behavior

The cross section behavior is shown in the figure:

0 0.5 1 1.5 2r (GeV)

0

10

20

30

σ dip

(x,r

) (m

b)GBW


Saturation and color transparency

• In GBW model, when r << 1/Q2s(Y ), the phenomenon of color transparency

appears (small dipoles have small chance of interaction):

σ(Y, r)

σ0=r2Q2

s(Y )

4.

• When r >> 1/Q2s(Y ), the cross section saturates:

σ(Y, r)

σ0≈ 1.


GBW Results

The γ∗p-cross section for various energies (solid lines: quark mass of 140MeV; dottedlines: zero quark mass).

W=245 (x128)

W=210 (x64)

W=170 (x32)

W=140 (x16)

W=115 (x8)

W=95 (x4)

W=75 (x2)

W=60 (x1)

ZEUS BPC

H1

H1 low Q 2

Q2 (GeV2)

σ γ*p (

µb)

Q2 (GeV2)

σ γ*p (

µb)

1

10

10 2

10 3

10 4

10-2

10-1

1 10 102


GBW Results

The results for the fit to the inclusive HERA data on F2 for different values of thevirtuality.

H1/ZEUS

F2

Q2=1.5 GeV2 Q2=2 Q2=2.5 Q2=3.5 Q2=4.5 Q2=5

Q2=6.5 GeV2 Q2=8.5 Q2=10 Q2=12 Q2=15 Q2=18

Q2=20 GeV2 Q2=22 Q2=25 Q2=27 Q2=35 Q2=45

Q2=60 GeV2 Q2=70 Q2=90 Q2=120 Q2=150 Q2=200

Q2=250 GeV2 Q2=350 Q2=450 Q2=650 Q2=800

H1(94/95)

ZEUS(94)

x

0

0.5

1

1.5

0

0.5

1

1.5

0

0.5

1

1.5

0

0.5

1

1.5

0

0.5

1

1.5

10-4

10-3

10-2

10-4

10-3

10-2

10-4

10-3

10-2

10-4

10-3

10-2

10-4

10-3

10-2

10-4

10-3

10-2


Saturation without unitarization

• Consider impact parameter dependence.

• Total inelastic cross section, in the saturation regime (t = ln(s/s0)):

σ = πR2(t) . (23)

• To satisfy the Froissart bound the radius R(t) should grow at most linearly with t.

• However, at large rapidities the radius of the saturated region is exponentially large

R(t) = R(t0) exphαsNc

2πǫ(t− t0)

i

. (24)

[A. Kovner, U. A. Wiedemann; Phys.Rev. D66 (2002) 051502.]


Saturation without unitarization


Geometric scaling

• Geometric scaling is a phenomenological feature of DIS which has been observedin the HERA data on inclusive γ∗ − p scattering, which is expressed as a scalingproperty of the virtual photon-proton cross section

10-1

1

10

10 2

10 3

10-3

10-2

10-1

1 10 102

103

E665

ZEUS+H1 high Q2 94-95H1 low Q2 95ZEUS BPC 95ZEUS BPT 97

x<0.01

all Q2

τ

σ totγ*

p [µ

b]

σγ∗p(Y,Q) = σγ∗p (τ) , τ =Q2

Q2s(Y )

where Q is the virtuality of the photon,Y = log 1/x is the total rapidity andQs(Y ) is the saturation scale

[Stasto, Golec Biernat and Kwiecinsky,2001]


Geometric scaling

• Looking to the solutions of BK equation, one sees that they reach a universalshape independently of the initial condition.

• The solutions even look similar when calculated for different Y (they appear to beshifted when Y is changed).

• In terms of the scattering amplitude, the geometric scaling means that BK solutiondepends only on rQs(Y ) for asymptotic values of rapidity.

N (r, Y ) = N (rQs(Y )).

• Using Qs(Y ) ≈ Q0exp(αsλY );

N (r, Y ) = N (Q0exp(lnr + αsλY ))

and the geometric scaling relation resembles as a traveling wave with velocity αsλ,time Y and spatial coordinate lnr.


Geometric scaling

√τσγ∗p plotted versus

the scaling variable τ .

10-1

1

10

10 2

10-3

10-2

10-1

1 10 102

103

E665

ZEUS+H1 high Q2 94-95

H1 low Q2 95ZEUS BPC 95ZEUS BPT 97

x<0.01

all Q2

τ

√τ– σto

tγ*p

[µb]


Geometric scaling

Different values of scal-ing variable versus ex-perimental data.

10-2

10-1

1

10

10 2

10-6

10-5

10-4

10-3

10-2

0.0075

0.0178

0.042

0.1

0.24

0.56

1.33

3.15

7.47 17.7 42.0 100

HERA fixed target

x

Q2 (G

eV2 )


Geometric scaling

Experimental data onσγ∗p from the regionx > 0.01 plotted ver-sus the scaling variableτ = Q2R2

0(x).

10-3

10-2

10-1

1

10

10 2

10-1

1 10 102

103

104

105

H1+ZEUS

BCDMSNMCE665EMCSLAC

x>0.01

all Q2

τ

σ totγ*

p [µ

b]


BK equation in momentum space

• Performing the Fourier transform:

N (k, Y ) =

Z

d2r

2πe−i~k·~r N (r, Y )

r2, (25)

BK equation is given by:

dN (k, Y )

dY= αs

Z

dk′

k′K(k, k′)N (k′, Y ) − αsN 2(k, Y ). (26)

• The solution to the linear part (BFKL) in saddle point approximation is:

kN (k, Y ) =1

p

παsχ′′(0)Yeαsχ(0)Y exp

− ln2(k2/k20)

2αsχ′′(0)Y

!

(27)

with χ(0) = 4 ln 2 and χ′′(0) = 28ζ(3).

• The last term in the expression above presents the diffusion.



0.5

1

1.5

2

2.5

3

10-5

10-4

10-3

10-2

10-1

1 10 102

103

104

BFKL

Kovchegov

k [GeV]

k φ(

k,y)



• BFKL presents strong diffusion and can be interpreted as a random walk in ln k

with rapidity as time.

• The Gaussian shapes of last figure are then expected, as well as the widthincrease of distributions.

• The initial condition used was

kN (k, Y = 0) = δ(k) (28)

• One sees that BK equation leads to a suppression of the diffusion into infrared.

• Therefore, one defines the saturation scale as the distribution peak positionQs(Y ) = kmax(Y ).

• Other way to see the same effects is using the distribution

Ψ(k, Y ) =kN (k,Y )

kmax(Y )N (kmax(Y ),Y ):



0

1

2

3

4

5

6

7

8

9

10

-5 -4 -3 -2 -1 0 1 2 3 4 5

αs = 0.2

Balitsky-Kovchegov

BFKL

log10(k/1GeV)

log 10

(1/x

)



• In the last figure, we see that BK equation shifts the contour plot towards highervalues of transverse momenta.

• Two regions are found: in the first, there is still diffusion, particularly for large k.

• In the second, diffusion is suppressed and the contour lines are parallel.

• This is an indication of geometric scaling, since straight lines can be parametrizedby ξ = ln k/k0 − λY + ξ0.

• One sees also that for large k BK and BFKL equations present similar solutions,but BFKL presents an unlimited increase with energy. On the other side, BKsolutions are bounded.


Traveling waves

• The BK equation can be compactly written in momentum space also as

∂Y N (ρ, Y ) = αχ(−∂ρ)N − αN 2. (29)

• The Mellin–transformed BFKL kernel χ(γ) is given by:

χ(γ) = 2ψ(1) − ψ(γ) − ψ(1 − γ), with ψ(γ) =Γ′(γ)Γ(γ)

. (30)

• χ(−∂ρ) is an integro-differential operator defined with the help of the formal seriesexpansion:

χ(−∂ρ) = χ(γ0)1 + χ′(γ0)(−∂ρ − γ01) +1

2χ′′(γ0)(−∂ρ − γ01)2

+1

6χ(3)(γ0)(−∂ρ − γ01)3 + . . . (31)

for some γ0 between 0 and 1, where we used the identity operator 1.


BK and FKPP equations

• Munier and Peschanski showed that after the change of variables

t ∼ αY, x ∼ log(k2/k20), u ∼ φ, (32)

• The approximation of BFKL kernel by the first three terms of the expansion:

χ(−∂ρ) = χ(γ0)1 + χ′(γ0)(−∂ρ − γ01) +1

2χ′′(γ0)(−∂ρ − γ01)2,

• And a saddle point approximation:

χ(−∂ρ) = −χ′(γc)∂ρ +1

2χ′′(γc)(−∂L − γc1)2,

• BK equation is reduced to the Fisher and Kolmogorov-Petrovsky-Piscounov(FKPP) equation from non-equilibrium statistical physics:

∂tu(x, t) = ∂2xu(x, t) + u(x, t) − u2(x, t), (33)

where t is time and x is the coordinate. FKPP dynamics is called reaction-diffusiondynamics.


Traveling waves

• FKPP equation admits so-called traveling wave solutions.

• The position of a wave front x(t) = v(t)t for a traveling wave solution does notdepend on details of nonlinear effects.

• At large times, the shape of a traveling wave is preserved during its propagation,and the solution becomes only a function of the scaling variable x− vt.

x

u(x, t)

0

1

t = 0 t1

X(t1)

2t1

X(t2)

3t1

X(t3)


Traveling waves and saturation

Y = 25Y = 20Y = 15Y = 10Y = 5Y = 0

log(k2)

T

4035302520151050-5

10

1

0.1

0.01

0.001

1e-04

1e-05

• In the language of saturation physics the position of the wave front is nothing butthe saturation scale x(t) ∼ lnQ2

s(Y ) and the scaling corresponds to the geometricscaling x− x(t) ∼ ln k2/Q2

s(Y ).

• Summarizing: time t→ Y ; space x→ L; wave front u(x− vt) → N (L− vY ) andtraveling waves → geometric scaling.


Traveling waves and saturation

• In the dilute regime, in which one has

T (k, Y )k≫Qs≈

„

k2

Q2s(Y )

«−γc

log

„

k2

Q2s(Y )

«

exp

"

− log2`

k2/Q2s(Y )

´

2αχ′′(γc)Y

#

, (34)

where

λ = min αχ(γ)

γ= α

χ(γc)

γc= αχ′(γc), α ≡ αsNc

π. (35)

• Geometric scaling is obtained within the window

log`

k2/Q2s(Y )

´

.p

2χ′′(γc)αY . (36)


Next-to-leading order BK

• As usual, to get the region of application of the leading-order evolution equationone needs to find the next-to-leading order (NLO) corrections.

• Unlike the DGLAP evolution, the argument of the coupling constant in LO BKequation is left undetermined in the LLA.

• Careful analysis of this argument is very important from both theoretical andexperimental points of view.

• Balitsky calculated the quark contribution to NLO B equation Phys.Rev. D75,014001 (2007).

• Balitsky and Chirilli calculated the gluon contribution to NLO B equation Phys.Rev.D77, 014019 (2008).



• NLO result does not lead automatically to the argument of coupling constant infront of the leading term.

• In order to get this argument, it was used the renormalon-based approach.

• The running coupling was roughly found to be αs(|x− y|), but:

αs((x−y)2)

2π2

(x−y)2

X2Y 2 |x− y| ≪ |x− z|, |y − z|αs(X2)

2π2X2 |x− z| ≪ |x− y|, |y − z|αs(Y 2)

2π2Y 2 |y − z| ≪ |x− y|, |x− z| (37)



• Next-to-leading order BK:

• X = x− z, Y = y − z, X = x− z′, Y = y − z′.

• As we can infer, in the complete hierarchy, the single dipole evolution is related tothe sextupole terms.


Beyond BK equation: Pomeron loops


Beyond BK equation: Fluctuations

• BK equation completely misses the effects of fluctuations in the gluon (dipole)number, related to discreteness of the evolution.

• Also, since BK equation uses a mean field approach, < T 2 > − < T >2= 0.

• After an approximation related to the impact parameter dependence, it can beshown that a Langevin equation for the event-by-event amplitude can rebuild thePomeron loop hierarchy.

• This is formally the BK equation with a Gaussian white noise term.

• It lies in the same universality class as the stochastic FKPP equation (sFKPP):

∂tu(x, t) = ∂2xu(x, t) + u(x, t) − u2(x, t) +

r

2

Nu(x, t)(u(x, t) − 1)ν(x, t). (38)

• White noise is defined as:

< ν(x, t) >= 0 < ν(x, t)ν(x′, t′) >= δ(t− t′)δ(x− x′). (39)

[E. Iancu, D.N. Triantafyllopoulos; Nucl.Phys.A 756, 419 (2005)]


Fluctuations

• Each realization of the noise means a single realization of the target in theevolution, which is stochastic, and this leads to an amplitude for a single event.

• Different realizations lead to a dispersion in the solutions and also in the saturationmomentum ρs ≡ ln(Q2

s(Y )/k20).

• For each single event, the evolved amplitude shows a traveling-wave pattern, withspeed of the wave smaller than the value predicted by BK equation:

λ∗ ≃ λ− π2γcχ′′(γc)

2 ln(1/α2s)

(40)

• The saturation scale Qs(Y ) is now a random variable whose average value isgiven by

〈Q2s(Y )〉 = exp [λ∗Y ]. (41)

• The dispersion in the position of the individual fronts is given by

σ2 = 〈ρ2s〉 − 〈ρs〉2 = DαY, (42)


Diffusive scaling

• Geometric scaling:

〈T (ρ, ρs)〉 = T (ρ− 〈ρs〉) . (43)

is replaced by diffusive scaling

〈T (ρ, ρs)〉 = T„

ρ− 〈ρs〉√αDY

«

. (44)


Fluctuations


Toy model• Are fluctuations really important (in the context of a toy model)?• At fixed coupling they are [E. Iancu, J.T. de Santana Amaral, G. Soyez, D.N.

Triantafyllopoulos; Nucl.Phys. A786, 131 (2007)].• However, at running coupling they are not [A. Dumitru, E. Iancu, L. Portugal, G.

Soyez, D.N. Triantafyllopoulos; JHEP 0708:062 (2007)].• This reflects the slowing down of the evolution by running coupling effects, in

particular, the large rapidity evolution which is required for the formation of thesaturation front via diffusion.

0

0.5

1

1.5

2

2.5

3

3.5

4

-10 0 10 20 30 40

T(x

-xs,

Y)

eγ s(x

-xs)

x-xs

MFA fluct.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-10 -5 0 5 10 15 20 25

T(x

-xs,

Y)

eγ s(x

-xs)

x-xs

MFA fluct.


References

Y.V. Kovchegov, Phys. Rev. D 60 (1999) 034008; Phys. Rev. D 61 (2000) 074018.I. Balitsky, Nucl. Phys. B 463 (1996) 99.A.M. Stasto, Acta Phys. Polonica 35 (2004) 3069.C. Marquet, G. Soyez, Nucl. Phys. A 760 (2005) 208.K. Golec-Biernat, M. Wüsthoff, Phys. Rev. D 59 (1999) 014017; Phys. Rev. D 60 (1999)114023; Eur. Phys. J. C 20 (2001) 313.S. Munier R.Peschanki, Phys. Rev. Lett. 91 (2003) 232001; Phys. Rev. D 69 (2004)034008; hep-ph/0401215.R.A. Fisher, Ann. Eugenics 7 (1937) 355. A. Kolnogorov, I. Petrovsky, N. Picounov,Moscow Univ. Bull. Math. A 1 (1937) 1.M.B. Gay Ducati, V.P. Goncalves, Nucl. Phys. B 557 296.A.M. Stasto, K. Golec-Biernat and J. Kwiecinsky, Phys. Rev. Lett. 86 (2001) 596;hep-ph/0007192.


AGL equation from BK equation

• AGL equation can be derived from BK equation. Taking the equation derived byKovchegov:

N (x01,b0, Y ) = −γ(x01,b0) exp

»

−4αsCF

πln

„

x01

ρ

«

Y

–

+αsCF

π

Z Y

0dy exp

»

−4αsCF

πln

„

x01

ρ

«

(y − Y )

–

×Z

ρd2x2

x201

x202x

212

[2N (x02,b0, Y −N (x02,b0, Y )N (x12,b0, Y )].(45)

• In the double logarithmic limit, in which the momentum scale of the photon Q2 islarger than the momentum scale of the nucleus ΛQCD, the large Q2 limit of lastequation reduces to:

∂N (x01,b0, Y )

∂Y=

αsCF

πx201

Z 1/Λ2QCD

x201

d2x02

x202x

212

[2N (x02,b0, Y

−N (x02,b0, Y )N (x02,b0, Y )]. (46)


AGL equation from BK equation

• The last equation can be derived again with respect to ln(1/x201Λ

2QCD):

∂N (x01,b0, Y )

∂Y ∂ ln(1/x201Λ2

QCD)=

αsCF

π[2 −N (x01,b0, Y )]N (x01,b0, Y ). (47)

• In the physical picture for the dipole evolution in the DLA limit, the produceddipoles at each step of the evolution have much greater transverse dimensionsthan the parent dipoles.

• The connection in this approximation between N and the nuclear gluon function is:

N (x01,b0 = 0, Y ) = 2

1 − exp

»

−αsCF π2

N2c S⊥

x201AxG(x, 1/x2

01)

–ff

. (48)

• From the above equations and using x01 ≈ 2/Q, one obtains:

∂N (x01,b0, Y )

∂Y=

αsCF

πx201

Z 1/Λ2QCD

x201

d2x02

x202x

212

[2N (x02,b0, Y

−N (x02,b0, Y )N (x02,b0, Y )]. (49)

• This equation is precisely the AGL equation.BK, E.G. de Oliveira–GFPAE – p. 67

Light-cone kinematics

• Let z be the longitudinal axis of the collision. For an arbitrary 4-vectorvµ = (v0, v1, v2, v3), the light-cone (LC) coordinates are defined as

v+ ≡ 1√2(v0 + v3), v− ≡ 1√

2(v0 − v3), v ≡ (v1, v2) (50)

• One usually writes v⊥ ≡ |v| =p

(v1)2 + (v2)2

• In these coordinates, x+ ≡ 1√2(t+ z) is the LC time and x− ≡ 1√

2(t− z) is the

LC longitudinal coordinate.

• The invariant scalar product of two 4-vectors reads

p · x = p0x0 − p1x1 − p2x2 − p3x3

=1

2(p+ + p−)(x+ + x−) − 1

2(p+ − p−)(x+ − x−) − p · x

= p−x+ + p+x− − p · x (51)

• This form of the scalar product suggests that p− should be interpreted as the LCenergy and p+ as the LC longitudinal momentum.


Light-cone kinematics

• For particles on the mass-shell, k± = (E ± kz)/√

2, with E2 = (m2 + k2z + k2)

k+k− =1

2(E2 − k2

z) =1

2(k2 +m2) ≡ m2

⊥ (52)

• One needs also the rapidity

y ≡ 1

2lnk+

k−=

1

2ln

2k+2

m2⊥

(53)

• Under a longitudinal Lorentz boost (k+ → βk+, k− → (1/β)k−, with constant β),the rapidity is shifted only by a constant, y → y + β

• For a parton inside a right-moving (in the positive z direction) hadron, we introducethe boost-invariant longitudinal momentum fraction x

x ≡ k+

P+(54)


balitsky–kovchegov evolution equation · introduction • when the center of mass energy in a...

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